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Complex brain networks: graph
theoretical analysis of structural and
functional systems
Ed Bullmore*‡ and Olaf Sporns§
Abstract | Recent developments in the quantitative analysis of complex networks, based
largely on graph theory, have been rapidly translated to studies of brain network organization.
The brain’s structural and functional systems have features of complex networks — such as
small-world topology, highly connected hubs and modularity — both at the whole-brain
scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we
review studies investigating complex brain networks in diverse experimental modalities
(including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the
basic principles of graph theory. We also highlight some of the technical challenges and key
questions to be addressed by future developments in this rapidly moving field.
Graph theory
A branch of mathematics that
deals with the formal
description and analysis of
graphs. A graph is defined
simply as a set of nodes
(vertices) linked by connections
(edges), and may be directed
or undirected. When describing
a real-world system, a graph
provides an abstract
representation of the system’s
elements and their interactions.
*University of Cambridge,
Behavioural & Clinical
Neurosciences Institute,
Department of Psychiatry,
Addenbrooke’s Hospital,
Cambridge, CB2 2QQ, UK.
‡
GlaxoSmithKline,
Clinical Unit Cambridge,
Addenbrooke’s Hospital,
Cambridge, CB2 2QQ, UK.
§
Department of Psychological
and Brain Sciences, Indiana
University, Bloomington,
Indiana 47405, USA.
Correspondence to E.B.
e‑mail: etb23@cam.ac.uk
doi:10.1038/nrn2575
Published online
4 February 2009
We have known since the nineteenth century that the
neuronal elements of the brain constitute a formidably
complicated structural network1,2. Since the twentieth
century it has also been widely appreciated that this
anatomical substrate supports the dynamic emergence
of coherent physiological activity, such as phase-locked
high-frequency electromagnetic oscillations, that can
span the multiple spatially distinct brain regions that
make up a functional network3,4. Such networks are
thought to provide the physiological basis for information processing and mental representations5–9. In this
article, we focus on graph theoretical approaches to the
analysis of complex networks that could provide a powerful new way of quantifying the brain’s structural and
functional systems (BOX 1).
Since the mid 1990s, developments in our understanding of the physics of complex systems10–12 have led
to the rise of network science13 as a transdisciplinary
effort to characterize network structure and function.
In this body of literature, complexity arises in the macroscopic behaviour of a system of interacting elements that
combines statistical randomness with regularity 14. The
ascendancy of network science has been driven by the
growing realization that the behaviour of complex systems — be they societies, cells or brains — is shaped
by interactions among their constituent elements. The
increasing availability and tractability of large, highquality data sets on a wide range of complex systems15–17
has led to a fundamental insight: substantively different
complex systems often share certain key organizational
principles, and these can be quantitatively characterized
by the same parameters (BOX 2). In other words, many
complex systems show remarkably similar macroscopic
behaviour despite profound differences in the microscopic details of the elements of each system or their
mechanisms of interaction.
One example of an apparently ubiquitous macroscopic behaviour in complex systems is the small-world
phenomenon18 (BOX 3). Recently, small-world architectures have been found in several empirical studies of
structural and functional brain networks19–22 in humans
and other animals, and over a wide range of scales in
space and time; small-worldness is naturally therefore
a key topic for this Review. However, discovering that
brain networks are small-world networks is only the
first step towards a comprehensive understanding of
how these networks are structurally organized and how
they generate complex dynamics. In network science,
methodological advances allow us to quantify other
topological properties of complex systems — such as
modularity 23, hierarchy 24, centrality 25 and the distribution of network hubs26,27 — many of which have already
been measured in brain networks. There have also been
significant efforts to model the development or evolution of complex networks28, to link network topology to
network dynamics, and to explore network robustness
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Box 1 | Structural and functional brain networks
Anatomical parcellation
Histological or
imaging data
2
Recording sites
1
2
3
Time series data
3
Structural brain network
Functional brain network
Sensorimotor
Premotor
Parietal
Prefrontal
Occipital
Inferior temporal
4
Orbitofrontal
Temporal pole
Graph theoretical analysis
Structural and functional brain networks can be explored using graph theory through the following four steps (see the figure):
•Define the network nodes. These could be defined as electroencephalography or multielectrode-array electrodes, or as
Nature Reviews | Neuroscience
anatomically defined regions of histological, MRI or diffusion tensor imaging data.
Complex network
An informal description of a
network with certain
topological features, such as
high clustering,
small-worldness, the presence
of high-degree nodes or hubs,
assortativity, modularity or
hierarchy, that are not typical
of random graphs or regular
lattices. Most real-life networks
are complex by this definition,
and analysis of complex
networks therefore forms an
important methodological tool
for systems biology.
Adjacency matrix
An adjacency matrix indicates
the number of edges between
each pair of nodes in a graph.
For most brain networks, the
adjacency matrix is specified
as binary — that is, each
element is either 1 (if there is
an edge between nodes) or 0
(if there is no edge). For
undirected graphs the
adjacency matrix is
symmetrical.
•Estimate a continuous measure of association between nodes. This could be the spectral coherence or Granger causality
measures between two magnetoencephalography sensors, or the connection probability between two regions of an
individual diffusion tensor imaging data set, or the inter-regional correlations in cortical thickness or volume MRI
measurements estimated in groups of subjects.
•Generate an association matrix by compiling all pairwise associations between nodes and (usually) apply a threshold to
each element of this matrix to produce a binary adjacency matrix or undirected graph.
•Calculate the network parameters of interest in this graphical model of a brain network and compare them to the
equivalent parameters of a population of random networks.
Each step entails choices that can influence the final results and must be carefully informed by the experimental question.
At step 1, parcellation schemes can use prior anatomical criteria or be informed by the functional connectivity profiles of
different regions. Several such parcellation schemes may be available and can affect network measures147. In most magnetoencephalography and electroencephalography studies, network nodes are equivalent to individual electrodes or sensors,
but networks could also be based on reconstructed anatomical sources. However, some reconstruction algorithms will
determine the brain location of each source by minimizing the covariance between sensors, which has major effects on the
configuration of functional networks. At step 2, a range of different coupling metrics can be estimated, including measures
of both functional and effective connectivity. A crucial issue at step 3 is the choice of threshold used to generate an
adjacency matrix from the association matrix: different thresholds will generate graphs of different sparsity or connection
density, and so network properties are often explored over a range of plausible thresholds. Finally, at step 4 a large number of
network parameters can be quantified (BOX 2). These must be compared with the (null) distribution of equivalent parameters
estimated in random networks containing the same number of nodes and connections. Statistical testing of network
parameters may best be conducted by permutation- or resampling-based methods of non-parametric inference given the
lack of statistical theory concerning the distribution of most network metrics.
Most graph theoretical network studies to date have used symmetrical measures of statistical association or functional
connectivity — such as correlations, coherence and mutual information — to construct undirected graphs. This approach
could be generalized to consider asymmetrical measures of causal association or effective connectivity — such as Granger
causal148,149 or dynamic causal66 model coefficients — to construct directed graphs. It is also possible to avoid the
thresholding step (step 3) by analysing weighted graphs that contain more information than the simpler unweighted and
undirected graphs that have been the focus of attention to date. Structural brain network image is reproduced from
ReF. 59. Functional brain network image is reproduced, with permission, from ReF. 70 © (2006) Society for Neuroscience.
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and vulnerability — topics that are likely to become
increasingly relevant in relation to neuroscience.
In this article, we describe and discuss the expanding
interface between systems neuroscience and the physics
of complex networks. We review the existing empirical
data on topological and dynamical properties of structural and functional brain networks, and ask what this
literature tells us about how structural networks shape
functional brain dynamics. Space limitations prevent us
from providing coverage of all animal models, in vitro
Box 2 | Network measures
A network is defined in graph theory as a set of nodes or vertices and the edges or lines between them. Graph topology
can be quantitatively described by a wide variety of measures, some of which are discussed here. It is not yet established
which measures are most appropriate for the analysis of brain networks. The figure shows a schematic diagram of a brain
network drawn as a directed (left) and an undirected (right) graph; both structural and functional networks can be either
directed or undirected (BOX 1).
Node degree, degree distribution and assortativity
The degree of a node is the number of connections that link it to the rest of the network — this is the most fundamental
network measure and most other measures are ultimately linked to node degree. The degrees of all the network’s nodes
form a degree distribution15. In random networks all connections are equally probable, resulting in a Gaussian and
symmetrically centred degree distribution. Complex networks generally have non-Gaussian degree distributions, often
with a long tail towards high degrees. The degree distributions of scale-free networks follow a power law90. Assortativity is
the correlation between the degrees of connected nodes. Positive assortativity indicates that high-degree nodes tend to
connect to each other.
clustering coefficient and motifs
If the nearest neighbours of a node are also directly connected to each other they form a cluster. The clustering coefficient
quantifies the number of connections that exist between the nearest neighbours of a node as a proportion of the
maximum number of possible connections18. Random networks have low average clustering whereas complex networks
have high clustering (associated with high local efficiency of information transfer and robustness). Interactions between
neighbouring nodes can also be quantified by counting the occurrence of small motifs of interconnected nodes150. The
distribution of different motif classes in a network provides information about the types of local interactions that the
network can support48.
Path length and efficiency
Path length is the minimum number of edges that must be traversed to go from one node to another. Random and
complex networks have short mean path lengths (high global efficiency of parallel information transfer) whereas regular
lattices have long mean path lengths. Efficiency is inversely related to path length but is numerically easier to use to
estimate topological distances between elements of disconnected graphs.
connection density or cost
Connection density is the actual number of edges in the graph as a proportion of the total number of possible edges and is
the simplest estimator of the physical cost — for example, the energy or other resource requirements — of a network.
Hubs, centrality and robustness
Hubs are nodes with high degree, or high centrality. The centrality of a node measures how many of the shortest paths
between all other node pairs in the network pass through it. A node with high centrality is thus crucial to efficient
communication151. The importance of an
individual node to network efficiency can be
assessed by deleting it and estimating the
efficiency of the ‘lesioned’ network.
Robustness refers either to the structural
integrity of the network following deletion of
nodes or edges or to the effects of
Threshold
perturbations on local or global network
states.
Modularity
Many complex networks consist of a number
of modules. There are various algorithms that
estimate the modularity of a network, many of
them based on hierarchical clustering23. Each
module contains several densely
interconnected nodes, and there are relatively
few connections between nodes in different
modules. Hubs can therefore be described in
terms of their roles in this community
structure27. Provincial hubs are connected
mainly to nodes in their own modules,
whereas connector hubs are connected to
nodes in other modules.
Motif
Path
Clustering
Connector hub
Community 2
Community 1
Provincial hub
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Box 3 | Random, scale-free and small-world networks
In random graphs each pair of nodes has an equal probability, p, of being connected152.
Large random graphs have Gaussian degree distributions (BOX 2). It is now known that
most graphs describing real-world networks significantly deviate from the simple
random-graph model.
Some networks (including the Internet and the World Wide Web) have degree
distributions in the form of a power law: that is, the probability that a node has degree k
is given as Prob(k) ~ k–λ. In biological systems, the degree exponent λ often ranges
between 2 and 3, and the very gradual (‘heavy-tail’) power law decay of the degree
distribution implies that the network lacks a characteristic scale — hence such
networks are called ‘scale-free’ networks. Barabási and Albert90 demonstrated that
scale-free networks can originate from a process by which each node that is added to
the network as it grows connects preferentially to other nodes that already have high
degree. Scale-free networks are unlikely if the attachment of connections is subject to
physical constraints or associated with a cost15. Therefore, physically embedded
networks, in which nodes have limited capacity for making connections, often do not
have pure power law degree distributions but may instead demonstrate exponentially
truncated power law degree distributions, which are associated with a lower
probability of very high degree nodes.
Originally described in social networks153, the ‘small-world’ property combines high
levels of local clustering among nodes of a network (to form families or cliques) and
short paths that globally link all nodes of the network. This means that all nodes of a
large system are linked through relatively few intermediate steps, despite the fact that
most nodes maintain only a few direct connections — mostly within a clique of
neighbours. Small-world organization is intermediate between that of random
networks, the short overall path length of which is associated with a low level of local
clustering, and that of regular networks or lattices, the high-level of clustering of which
is accompanied by a long path length18. A convenient single-number summary of
small-worldness is thus the ratio of the clustering coefficient to the path length after
both metrics have been standardized by comparing their values to those in equivalent
random networks154. Evidence for small-world attributes has been reported in a wide
range of studies of genetic, signalling, communications, computational and neural
networks. These studies indicate that virtually all networks found in natural and
technological systems have non-random/non-regular or small-world architectures
and that the ways in which these networks deviate from randomness reflect their
specific functionality.
Connectivity
In the brain, connectivity can
be described as structural,
functional or effective.
Structural connectivity denotes
a network of anatomical links,
functional connectivity denotes
the symmetrical statistical
association or dependency
between elements of the
system, and effective
connectivity denotes directed
or causal relationships
between elements.
Microcircuit
A neuronal network composed
of specific cell types and
synaptic connections, often
arranged in a modular
architecture and capable of
generating functional outputs.
Connectome
The complete description of
the structural connections
between elements of a nervous
system.
preparations and human studies that have contributed to
this endeavour: we necessarily focus on what we consider
to be some representative examples of graph theoretical
research in brain networks, with an emphasis on studies
of the human brain. We thus consider the implications of
complex brain networks for an understanding of neuropsychiatric disorders and conclude with some general
remarks about the evolution of scale-invariant topology
in brain networks and some key future challenges for this
emerging field.
Structural brain networks
Topological and physical properties of structural networks. The anatomical configuration of brain networks,
ranging from inter-neuronal connectivity to inter-regional
connectivity, has long been a focus of empirical neuroscience. network analysis, and in particular graph theory
(BOX 4), offers new ways to quantitatively characterize
anatomical patterns. according to graph theory, structural brain networks can be described as graphs that are
composed of nodes (vertices) denoting neural elements
(neurons or brain regions) that are linked by edges representing physical connections (synapses or axonal
projections). although graph theory emphasizes topological connectivity patterns, the topological and physical
distances between elements in brain networks are often
intricately related29. neurons and brain regions that are
spatially close have a relatively high probability of being
connected, whereas connections between spatially remote
neurons or brain regions are less likely 30–32. longer axonal
projections are more expensive in terms of their material
and energy costs33. It has been suggested that the spatial
layout of neurons or brain regions34,35 is economically
arranged to minimize axonal volume. Thus, conservation of wiring costs is likely to be an important selection
pressure on the evolution of brain networks.
Mapping structural networks in animal models. Tracing
individual neuronal processes has been a long-standing
technological challenge, and few structural brain networks have been mapped at cellular resolution. Serial
reconstruction of electron microscopy sections allowed
the complete connection matrix of the nematode
Caenorhabditis elegans to be visualized36. currently,
this is the only nervous system to have been comprehensively mapped at a cellular level, and it was the first
to be described as a small-world network18. In larger
brains, combinations of physiological and anatomical
techniques have allowed patterns in neuronal connections to be identified, leading to the identification of
neuronal microcircuits37 and the formulation of probabilistic connection rules30. Small-world properties have
been demonstrated in biologically accurate models of
cellular networks in the reticular formation of the vertebrate brainstem38. Reconstruction of cellular networks in
the mammalian neocortex from multielectrode activity
recordings has revealed several highly nonrandom features of connectivity 39, including a tendency for synaptic
connections to be reciprocal and clustered. a promising
approach for mapping connectivity involves the stochastic expression of several fluorescent proteins40, and this
may ultimately deliver a complete map of the cellular
interconnections of an entire brain41.
Histological dissection and staining, degeneration
methods and axonal tracing have been used to map cerebral white matter connections. The pathways identified by these methodologies have formed the basis for
the systematic collation of species-specific anatomical
connection matrices, including those for the macaque
visual cortex 42 and the cat thalamocortical system43.
network analyses of such data sets demonstrated high
clustering of functionally related areas with short average
path lengths44–46, hallmarks of a small-world architecture.
clusters identified by network analysis map on to known
functional subdivisions of the cortex 43. long-distance
cortical projections facilitate short-path communication despite increasing axonal volume47. The cortical
networks of several mammalian species also consistently
demonstrated an overabundance of motif classes associated with network modularity and functionally diverse
circuitry 48,49.
Mapping structural networks in the human brain.
Several attempts have been made to map the structural networks of the human brain, also known as the
human connectome50, at the scale of brain regions. One
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Box 4 | Origins of graph theory
Diffusion tensor imaging
(DTI). An MRI technique that
takes advantage of the
restricted diffusion of water
through myelinated nerve
fibres in the brain to map the
anatomical connectivity
between brain areas.
Diffusion spectrum imaging
An MRI technique that is
similar to DTI, but with the
added capability of resolving
multiple directions of diffusion
in each voxel of white matter.
This allows multiple groups of
fibres at each location,
including intersecting fibre
pathways, to be mapped.
Cortical parcellation
A division of the continuous
cortical sheet into discrete
areas or regions; Brodmann’s
division of the cortex into areas
defined by their
cytoarchitectonic criteria is the
most famous but not the only
parcellation scheme.
Neuronographic
measurements
Recordings of epileptiform
electrical activity at specific
sites in the cortex following
topical application of a
pro-convulsive drug to a distant
cortical site; rapid propagation
of electrical activity from
stimulation to recording sites
implies that the sites are
anatomically connected.
Functional MRI
(fMRI). The detection of
changes in regional brain
activity through their effects on
blood flow and blood
oxygenation (which, in turn,
affect magnetic susceptibility
and tissue contrast in magnetic
resonance images).
Electroencephalography
(eeG). A technique used to
measure neural activity by
monitoring electrical signals
from the brain, usually through
scalp electrodes. eeG has good
temporal resolution but
relatively poor spatial
resolution.
Magnetoencephalography
(MeG). A method of measuring
brain activity by detecting
minute perturbations in the
extracranial magnetic field that
are generated by the electrical
activity of neuronal
populations.
Graph theory is rooted in the physical world155. In 1736, Euler showed that it was impossible to traverse the city of
Königsberg’s seven bridges across the river Pregel exactly once and return to the starting point. To prove this conjecture,
Euler represented the problem as a graph, and his original publication156 is generally taken to be the origin of a new
branch of mathematics called graph theory. In the middle of the nineteenth century, the analysis of electrical circuits and
the exploration of chemical isomers led to the discovery of additional graph theoretic concepts149. Today, graph theory
pervades many areas of science.
Significant progress in graph theory has come from the study of social networks157. One prominent experiment153
tracked paths of acquaintanceship across a large social network and found that even very large networks could be
traversed, on average, in a small number of steps. Although this ‘small-world’ phenomenon quickly captured the public
imagination, its origins remained obscure until its association with specific types of connectivity was demonstrated
(BOX 3). The dual discoveries of small-world18 and scale-free90 networks launched the modern era of graph theory, which
now extends into biology and neuroscience. Neural systems have long been described as sets of discrete elements linked
by connections. Nonetheless, graph theory has essentially only been applied to neuroscience in the past 10 years.
study derived structural connection patterns from crosscorrelations in cortical thickness or volume across
individual brains, which might indirectly indicate the
presence of corticocortical pathways51,52. Graph analysis
revealed small-world attributes and the existence of local
communities of brain regions. a more detailed analysis53
of the modularity or community structure of connection
data sets derived from cortical thickness correlations
revealed significant overlap between anatomical network
modules and functional systems in the cortex.
Human brain structural networks have also been
mapped using diffusion imaging and tractography. a map
of 70–90 cortical and basal brain grey matter areas was
constructed using diffusion tensor imaging (DTI) and analysed using methods from graph theory 54,55. The network
exhibited high clustering and short path length, and contained motif classes similar to those identified from tracttracing data48. Several areas, including the precuneus, the
insula, the superior parietal cortex and the superior frontal
cortex, were found to have high ‘betweenness centrality’
and thus to constitute putative hubs. another study that
mapped connections between 78 cortical regions using
DTI also identified several hub regions, including the
precuneus and the superior frontal gyrus56.
Due to limitations in the model that is used to infer
fibre bundle orientation, DTI has difficulty detecting
crossing fibre bundles. Diffusion spectrum imaging can overcome this limitation by reconstructing multiple diffusion
directions in each voxel57 and was used to build cortical
connection matrices between 500–4,000 homogeneously
distributed regions of interest 58. again, network analyses
revealed the small-world architecture of the cortical network. a more extensive analysis of 998 region-of-interest
networks obtained from 5 participants 59 identified
structural modules interconnected by highly central hub
regions. When considering multiple network measures
(including node degree, connection strength and centrality), a particular set of brain regions located predominantly in the posterior medial cortex, including portions
of the posterior cingulate and the precuneus, was highly
and densely interconnected, forming a structural core59.
although they differ in terms of their experimental methodology and cortical parcellation, most of these
studies reveal highly clustered large-scale cortical
networks, with most pathways existing between areas
that are spatially close and functionally related. These
clusters or modules are interlinked by specialized hub
regions, ensuring that overall path lengths across the
network are short. most studies identified hubs among
parietal and prefrontal regions, providing a potential
explanation for their well-documented activation
by many cognitive functions. Particularly notable is
the prominent structural role of the precuneus 55,56,59,
a region that is homologous to the highly connected
posteromedial cortex in the macaque60. The precuneus
is involved in self-referential processing, imagery and
memory 61, and its deactivation is associated with anaesthetic-induced loss of consciousness62. an intriguing
hypothesis suggests that these functional aspects can
be explained on the basis of its high centrality in the
cortical network.
Functional brain networks
although analysing structural networks helps us to
understand the fundamental architecture of interregional connections, we must also consider functional
networks directly to elucidate how this architecture supports neurophysiological dynamics. Despite considerable
heterogeneity in the methodological approaches, there is
an encouraging degree of convergence between studies
of functional brain networks. The first such study used a
set of neuronographic measurements of the propagation of
epileptiform activity following localized applications
of strychnine to the macaque cortex 63. This demonstrated a pattern of functional connections between
cortical areas that was consistent with a small-world
network. as we discuss below, these findings have been
extended by studies based on functional MRI (fmRI),
electroencephalography (eeG), magnetoencephalography
(meG) or multielectrode array (mea) data.
although such studies based on graph theory are
the focus of this Review, we note that other methods to
investigate brain functional systems have recently been
developed, including mathematical models of effective connectivity between regions. effective connectivity models, such as structural equation modelling 64,65,
dynamic causal modelling 66 or Granger causality 67,
involve estimating the causal influence that each element of a system exerts on the behaviour of other
elements. Thus, measures of effective connectivity
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between multiple regions can be used to generate a
directed graph, which can then be topologically described
using graph theory. However, the functional network
studies reviewed below have all been based on undirected graphs, derived from simpler measures of functional connectivity or symmetrical statistical association
between brain regions. The key point is that, in principle,
graph theory could be applied to an association matrix of
either functional or effective connectivity measures, to
generate either undirected or directed graphs, respectively, although all neuroscientific studies to date have in
fact been based on measures of functional connectivity.
Multielectrode array
(MeA). A technique for
simultaneously measuring the
electrical activity of local
neuronal populations or single
neurons, usually in tissue slices
or cell cultures in vitro.
Association matrix
A matrix that represents the
strength of the association
between each pair of nodes in
a graph. Association between
nodes can be quantified by
many continuously variable
metrics, such as correlation or
mutual information. either
functional or effective
connectivity measures can be
used to construct an
association matrix.
Blood oxygen
level-dependent (BOLD)
signals
Changes in magnetic
susceptibility and MRI tissue
contrast that are indirectly
indicative of underlying
changes in spontaneous or
experimentally controlled brain
activation.
Default-mode network
A set of brain regions, including
medial frontal and posterior
cingulate areas of the cortex,
that are consistently
deactivated during the
performance of diverse
cognitive tasks.
Mapping functional networks using fMRI. The first
graph theoretical study of fmRI data measured the
partial correlations of resting-state blood oxygen leveldependent (BOLD) signals between 90 cortical and subcortical regions and reported small-world properties of
the resulting whole-brain networks68. almost simultaneously, another study reported small-world properties
of functional networks derived from a set of activated
voxels in fmRI data; this voxel-level network was also
reported to have a scale-free degree distribution69.
Subsequently, small-world properties, with parameter
values similar to those previously reported in topological studies of cat and macaque anatomical connectivity
matrices50, were confirmed in a low-frequency (0.03–
0.06 Hz) whole-brain network derived from wavelet
correlations between regional mean time series70. The
high-degree nodes or hubs of this network were mostly
regions of multimodal association cortex, and the degree
distribution was an exponentially truncated power law 70.
Other studies have explored the community structure of
fmRI networks using a hierarchical cluster analysis68,71,72
and shown that functionally and/or anatomically related
brain regions are more densely interconnected, with relatively few connections between functional clusters, again
echoing prior work on anatomical connectivity matrices.
The high density of connections between functionally
related regions increases the clustering coefficient of
the graph, whereas the long-range connections between
different modules or clusters, even though they are relatively few in number, keep the path length low. Thus, the
small-world architecture of a brain functional network
is closely related to its modularity 72.
There are several other metrics for quantifying smallworld architecture in brain functional networks. Studies
in statistical physics73,74 have shown that path length is
inversely related to the global efficiency of a network for
the transfer of information between nodes by multiple
parallel paths, and that global efficiency is easier to estimate than path length when studying sparse networks.
Furthermore, the clustering coefficient can be regarded as
a measure of the local efficiency of information transfer,
or of the robustness of the network to deletion of individual nodes. The structural network of the macaque brain
was found to have high global and local efficiency and to
be sparsely connected73,74. Thus, the macaque cortex has
‘economical small-world’ properties: it has high global
efficiency of parallel information transfer and high local
fault tolerance for relatively low connection density.
These concepts were translated to the analysis of
resting-state fmRI data acquired from young and elderly adults75, using the wavelet correlation (a measure
of the association between time series in a specific frequency band) to estimate the functional connectivity
between regional BOlD time series endogenously oscillating in the frequency interval 0.06–0.1 Hz (for a more
detailed review of the rationale for wavelet analysis in
fmRI, see ReF. 76). In younger adults, functional brain
networks demonstrated small-world properties over a
broad range of connection densities or ‘costs’. Relatively
sparse networks were associated with maximum cost
efficiency. The older age group also showed evidence of
small-world properties, but had significantly reduced
cost efficiency: they had to be relatively over-connected
to provide the efficiency of parallel information transfer seen in a younger brain network. The suggestion
that aging is associated with changes in the economical small-world properties of brain functional networks
converges with studies that show differences in attentional and default-mode networks between children and
young adults77. normal processes of brain maturation
and senescence might thus be reflected in quantifiable
changes in functional network topology.
Mapping functional networks using electrophysiological techniques. When comparing the results of fmRI
studies to results obtained using electrophysiological
techniques (eeG, meG or mea), many aspects of the
data clearly differ. fmRI has good spatial resolution (on
the order of millimetres) but poor temporal resolution
(on the order of seconds), restricting the measurable
bandwidth to approximately 0.001–0.5 Hz, and fmRI
measures activation-related haemodynamics rather than
neuronal activity per se. all electrophysiological methods measure neuronal activity more directly and have
better temporal resolution, with bandwidths typically of
1–100 Hz, but they often have worse spatial resolution
(on the order of millimetres or centimetres for meG and
eeG) or less complete anatomical coverage (in the case
of mea) than fmRI. another point of difference is that
the nodes of a network derived from fmRI data will be
anatomically localized regions or voxels of the image,
whereas the nodes of a network derived from meG or
eeG data could be the surface sensors or recording electrodes. However, we can compare the topologies of networks derived from these different data sets by invoking
a general operating principle of complex network analysis: microscopically distinct systems can be informatively
compared in terms of their macroscopic organization
using graph theory.
Functional connectivity between pairs of electrodes
has been estimated using a measure of generalized synchronization78, and then thresholded to generate functional networks, in several studies of eeG or meG data
sets. This has shown that small-world topology is represented at many frequency intervals79 and can be related
to cognitive performance and normal aging 80. an alternative approach used the wavelet correlation to estimate
frequency-dependent functional connectivity between
meG sensors, again revealing that many topological and
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dynamic properties of brain functional networks were
conserved across frequencies81. This timescale invariance of wavelet correlations and the brain functional
network parameters derived from them is a theoretically
predictable corollary of the long-range autocorrelations
of neurophysiological time series82–84.
Metastable dynamics
Transitions between marginally
stable network states; these
transitions can occur
spontaneously or as a result of
weak external perturbations.
Resting state
A cognitive state in which a
subject is quietly awake and
alert but does not engage in or
attend to a specific cognitive
or behavioural task.
Conservation of functional network properties. There
have been fewer studies of functional networks in nonhuman species. In anaesthetized rats, fmRI has been
used to demonstrate small-world and modular properties of whole-brain functional networks85. a recent
study considered functional coupling between pairs of
cortical neurons, measured by multiunit electrodes, in
the visual cortex of the anaesthetized cat 86. most of the
properties of these neuronal-interaction networks could
be accounted for by the pairwise correlations between
electrodes87,88, and the networks had small-world properties with some highly connected (hub) neurons. The
small-world organization of functional networks at a cellular level has also been described on the basis of mea
recordings from in vitro cultures of cortical networks89.
These results indicate that small-worldness might be a
conserved property of brain functional networks over
different species and spatial scales.
Preliminary evidence suggests that other topological properties of brain functional networks, such as the
degree distribution, might also be conserved over spatial
and temporal scales. many large networks exhibit scalefree power law degree distributions90 indicative of the
existence of highly connected nodes (BOX 3). Some studies of functional brain networks carried out at high spatial
resolution (single voxels in fmRI) have provided evidence
of scale-free organization69,91. However, in other studies of
functional networks derived from fmRI and meG data
sets over a wide range of frequency intervals70,81, the
empirical degree distribution conforms to an exponentially truncated power law, implying that the probability of a highly connected node or hub is greater than
in an equivalent random network but less than would
be expected in a scale-free network. Truncated power
law degree distributions have also been reported from
analysis of anatomical networks derived from structural
mRI data in humans55, and from analysis of functional
networks derived from mea data in cats86. These distributions are typical of physically embedded networks,
such as the global air-transportation network, in which
the maximum degree is limited by physical considerations such as the finite capacity of any node to receive
connections27. The reasons for the differences between
these findings and reports of scale-free properties are
currently unknown; however, it is notable that pure
power law scaling of the degree distributions of human
brain functional networks has only been reported by
voxel-level analysis, whereas exponentially truncated
power laws have been reported by region-level analysis.
The form of the degree distribution could be affected
by spatial interpolation, and other pre-processing steps
applied before the construction of functional networks
and the standardization of such methodological procedures will be important in elucidating the impact of
anatomical resolution on the extent to which power
law scaling of the degree distribution is truncated at
high degree.
Structure–function relations in brain networks
How do functional brain networks emerge from structural brain connectivity? Structural maps indicate that
each neural node maintains a specific pattern of structural
connections with other nodes. Different nodes often have
different functionalities, such as specific response preferences to sensory stimuli. From a network perspective, the
functionality of an individual neural node is partly determined by the pattern of its interconnections with other
nodes in the network92: nodes with similar connection
patterns tend to exhibit similar functionality 36. although
functional properties are expressed locally, they are the
result of the action of the entire network as an integrated
system. Structural connectivity places constraints on
which functional interactions occur in the network.
Structural and functional connectivity in cellular networks undergo dynamic changes. The degree to which
synaptic connectivity is modified in the adult brain is
highly debated. although some evidence suggests that
mammalian cellular networks are continually remodelled93, other evidence indicates that most synaptic
spines are stable94. changes in neuronal connectivity
necessitate homeostatic mechanisms to ensure functional stability 95. multielectrode recording data suggest
that cellular functional networks exhibit transient synchronization96 and metastable dynamics97. These changes
occur within seconds, and it seems possible that even
more rapid transitions and network reconfigurations
may take place33. These observations of relatively slow
structural modifications accompanied by faster changes
in functional linkages pose major unresolved questions
regarding functional stability in neural circuits.
It is currently unknown whether large-scale cortical
networks in the adult brain undergo structural modifications on fast timescales. most of the changes that
have been observed were associated with aging, disease
progression or experience-dependent plasticity. By contrast, patterns of functional connectivity between brain
regions undergo spontaneous fluctuations and are
highly responsive to perturbations, such as those that
are induced by sensory input or cognitive tasks, on a
timescale of hundreds of milliseconds. These rapid reconfigurations do not affect the stability of global topological characteristics81,98. On longer timescales of seconds
to minutes, correlations between spontaneous fluctuations in brain activity 99–101 form functional networks that
are particularly robust. For example, a set of posterior
medial, anterior medial and lateral parietal brain regions
comprises the default mode network102,103.
The persistence of functional networks associated
with the brain’s resting state provides an opportunity
to investigate how much of the pattern of functional
connections is determined by underlying structural
networks. Observations from a single cortical slice104,
structural imaging of fibre bundles linking components
of the default network105, and direct comparisons of
structural and functional connectivity in the same cohort
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Structural brain network
Anatomical
tract tracing
Coupling matrix and neural dynamics
Simulated
electrophysiological
time series
Simulated BOLD
time series
Functional brain network
Mutual information
Cross-correlation
Figure 1 | computational modelling of structural and functional brain networks.
Computational models have been used to demonstrate how dynamic patterns arise as a
Nature Reviews | Neuroscience
result of interactions between anatomically connected neural units. Shown is how such
a model is generated and used. A structural brain network derived from anatomical data
serves as a matrix of coupling coefficients that link neuronal nodes, the activities of which
unfold through time. This time evolution is governed by physiologically motivated dynamic
equations. In the example shown, the surface of the macaque cortex was subdivided into
47 areas (nodes) and a structural brain network linking these nodes was compiled from
anatomical tract-tracing data. The dynamic equations were derived from a model of large
neuronal populations, the parameters of which were set to physiological values109. Data
from computer simulations then yield functional brain networks. Such networks are
derived from measures of association between the simulated time series — for example, an
information theoretic measure such as the mutual information (computed on
voltage–time data) or cross-correlations in neural activity that are computed from
simulated blood oxygen level-dependent (BOLD) data. These matrices can then be
thresholded to yield binary networks from which network measures can be derived. The
fact that both structural and functional networks are completely specified in the model
facilitates their comparative analysis. The structural brain network panel is reproduced,
with permission, from ReF. 109  (2007) National Academy of Sciences. The rest of the
figure is modified, with permission, from ReF. 158  (2009) Academic Press.
of participants63,106,107 suggest that structural connections
are highly predictive of functional connections. Indirect
interactions can account for additional functional linkages. Such indirect connections can lead to discrepancies between structural and functional connectivity;
however, current evidence suggests that topological
parameters are generally conserved between structural
and functional networks.
computational models offer a complementary
method to investigate structure–function relations in
brain networks. Several studies have used empirically
derived structural brain networks to specify the wiring
between neural units, which then results in spatially
patterened dynamic interactions (FIG. 1). The covariance structure of the endogenous activity generated by
these neural units yields a functional network that can
be generated by computer simulations or in some cases
analytically 108. This strategy has been used to determine
the effects of structural topology on functional networks
and dynamics. Simulation studies of large-scale cortical networks demonstrated the emergence of complex
spatiotemporal structure in neural correlations at multiple timescales109, as well as realistic patterns of modelled BOlD resting-state correlations that depend on
the topology 109 and time delays110 in the structural coupling matrix. The topology of structural and functional
networks was identical when functional connectivity
was estimated from long time samples, but functional
networks estimated on shorter time samples or at higher
frequencies were less strongly constrained by the structural wiring diagram109. likewise it has been shown
that the modularity of structural networks can determine the hierarchical organization of functional networks111,112 and may be important for generating diverse
and persistent dynamic patterns113. In a computational
model of phase synchronization between coupled neurons, patterns of local versus global synchronization
were found to depend strongly on the balance between
high clustering and short path length in the connectivity of the network114. computational studies have also
begun to determine the effects of functional activity on
structural topology. It has been shown that an initially
random wiring diagram can evolve through synaptic
plasticity to a functional state characterized by a smallworld topology of the most strongly connected nodes
and by self-organized critical dynamics115.
These findings indicate that the brain’s structural
and functional networks are intimately related and
share common topological features, such as modules
and hubs (FIG. 2). although most studies provide support for the idea that structural networks determine
some aspects of functional networks, especially at low
frequencies or over long time periods, it is less clear how
the structural topology both supports the emergence of
fast and flexibly reconfigured functional networks and
is itself remodelled by function-related plasticity on a
slower timescale.
Clinical and translational aspects
Since the work of pioneers such as Wernicke, meynert
and Dejerine, it has been appreciated that many neurological and psychiatric disorders can be described as dysconnectivity syndromes116. The emergence of symptoms
or functional impairment in these disorders can be theoretically related to the disruption or abnormal integration
of spatially distributed brain regions that would normally
constitute a large-scale network subserving function.
Using network properties as diagnostic markers. One
application of complex-network theory in this context is
to provide new measures to quantify differences between
patient groups and appropriate comparison groups.
Several studies have reported that the parameters of
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Cellular functional network
Module 1
Hub node
Module 2
PIP
46
VP
DP
CITd
VOT
VIP
LIP
MT
V2
PO
V4
V3
PITv
TF
FEF
AITv
V3A
V1
TH
FST
V4t
Whole-brain structural network
PITd
CITv
Figure 2 | cellular and whole-brain networks demonstrate consistent topological
Reviews
| Neuroscience
features. The top panel shows a cellular functional networkNature
constructed
from
multielectrode-array recordings made in the anaesthetized cat; each node (represented by a
circle) corresponds approximately to one neuron and the connections represent high
functional connectivity between neurons86. The different coloured nodes constitute
separate clusters or modules. The plots in each circle illustrate cellular responses to
stimuli of different orientations, and the circle size corresponds to the degree (number of
functional connections) of each node. The bottom panel shows a whole-brain structural
network constructed from histological data on the macaque cortex; each node
corresponds to a brain area and the connections represent axonal projections between
areas49. The network has two main modules, shown here with yellow and grey circles
corresponding to mostly dorsal and ventral visual regions, respectively. Both networks
exhibit the small-world attributes of high clustering and short path length (see BOX 3);
both have an exponentially truncated power law degree distribution (see BOX 2),
associated with the existence of high-degree ‘hubs’ (V4 in the anatomical network); and
both have a community structure characterized by sparse connectivity between modules
(each module is enclosed by stippled lines) and linked by hubs (nodes circled in red).
AITv, anterior inferotemporal ventral area; CITd, central inferotemporal dorsal area;
CITv, central inferotemporal ventral area; DP, dorsal preluneate area; FEF, frontal eye
field; FST, floor of superior temporal area; LIP, lateral intraparietal area; MT, middle
temporal area; PIP, posterior intraparietal area; PITd, posterior inferotemporal dorsal
area; PITv, posterior inferotemporal ventral area; PO, parieto-occipital area; TF, area TF;
TH, area TH ;V1–4, visual cortical areas 1–4; VIP, ventral intraparietal area; VOT, ventral
occipitotemporal area; VP, ventral posterior area. The top panel is reproduced, with
permission, from ReF. 86  (2008) Oxford University Press. The bottom panel is
reproduced from ReF. 49.
brain networks derived from fmRI, eeG or structural
mRI data are altered in patients with schizophrenia or
alzheimer’s disease (aD).
In an fmRI study of aD, clustering was significantly
reduced at a global level (in whole-brain networks
operating at frequencies below 0.1 Hz) and at a local
level (in both hippocampi), and global clustering was
able to discriminate aD patients from age-matched
comparison subjects with high specificity and sensitivity, implying that the loss of small-world network
properties might provide a clinically useful diagnostic
marker 117. In a comparable eeG study, path length in
beta band (15–35 Hz) functional networks was significantly increased in patients with aD. Importantly, the
variability of cognitive function across both control and
aD groups was negatively correlated with path length,
providing direct evidence that functional-network
topology can be related to variation in cognitive performance118. a third meG study of resting-state functional networks confirmed a degradation of small-world
attributes in patients with aD and suggested that this
effect is due to disease-related changes at highly connected network hubs119. However, in a fourth study that
used between-subject covariation in regional measures
of cortical thickness to infer anatomical networks from
a large structural mRI data set, global clustering was
significantly increased in patients with aD and there
were abnormalities in the topological configuration
of crucial, high-centrality nodes in regions of the
multimodal association cortex 120.
Schizophrenia has been investigated by comparable
approaches. The economical small-world properties of
low-frequency functional networks derived from fmRI
data were shown to be impaired in patients with schizophrenia121. Studies that used eeG to measure synchronization likelihood or nonlinear interactions between
cortical nodes found that the clustering and path-length
parameters of functional networks were closer to their
values in random graphs for patients with schizophrenia
than for comparison subjects122,123. anatomical networks
derived from inter-regional covariation of grey matter density in structural mRI data showed differences
in the hierarchy and assortativity of multimodal and
transmodal cortical subnetworks in healthy volunteers,
suggesting that these major divisions of the cortex may
have developed according to different growth rules or
evolved to meet different selection criteria. Topological
abnormalities in people with schizophrenia included
a reduced hierarchy of multimodal cortex (FIG. 3). The
schizophrenic group’s networks also exhibited relatively
long physical distances between connected regions,
compatible with inefficient axonal wiring 124.
Thus, there is convergent evidence from methodologically disparate studies that both aD and schizophrenia are associated with abnormal topological
organization of structural and functional brain networks. However, there are also inconsistencies between
existing studies — for example, clustering is reportedly
increased in the structural networks120 but decreased in
the functional networks of patients with aD117 — which
might be attributable to the clinical heterogeneity of the
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a Healthy volunteers
b People with schizophrenia
37
Frontal
Temporal
Parietal
Occipital
High clustering
Low clustering
44'
45'
46
12
37
20
36'
Degree
8
22
34'
44
21'
19
36
46'
47'
9
44
22
21
10
43'
46
39
43
39'
22
42'
47'
40
47
20
8'
9'
8
40
44'
10'
40'
20'
43
45'
39'
45
10'
20
37'
39
40'
19'
37'
19'
47
10
8'
45
36
43'
19
9'
21'
9
22'
Figure 3 | Disease-related disorganization of brain anatomical networks derived
from structural Mri data. In both parts, the nodes (circles) represent cortical regions
Nature
Reviews
| Neuroscience
and the connections represent high correlation in grey matter
density
between
nodes.
The nodes are arranged vertically by degree and are separated horizontally for clarity of
representation. The numbers indicate approximate Brodmann area, and the prime
symbols (′) denote left-sided regions. The clustering coefficient of each node, a measure
of its local connectivity, is indicated by its size: nodes with high clustering are larger.
a | The brain anatomical network of the healthy volunteers has a hierarchical
organization characterized by low clustering of high-degree nodes24. b | The equivalent
network constructed from MRI data on people with schizophrenia shows loss of this
hierarchical organization — high-degree nodes are more often highly clustered. Figure is
reproduced, with permission, from ReF. 124  (2008) Society for Neuroscience.
Assortativity
A measure of the tendency for
nodes to be connected to
other nodes of the same or
similar degree.
Endophenotype
A quantifiable biological
marker of the genetic risk for a
neuropsychiatric disorder.
patient groups as well as to the differences in imaging and analytic methods. Some of these differences
may perhaps be resolved by studies combining network
measurements on structural and functional neuroimaging data acquired on the same patients. It also seems
likely that evidence for network abnormalities in other
neuropsychiatric disorders and conditions (such as epilepsy 125–127, attention-deficit hyperactivity disorder 128 or
spinal cord injury 129) will accumulate as the disorders
are increasingly investigated from this perspective.
Understanding the pathogenesis and treatment of brain
disorders from a network perspective. many psychiatric
disorders are highly heritable and are likely to represent the clinical outcome of aberrations in the formation of large-scale networks in utero or during early
postnatal life. measures of network topology may be
worth investigating as intermediate phenotypes, or
endophenotypes , that indicate the genetic risk for a
neuropsychiatric disorder; however, network metrics
have not yet been adopted for this purpose. a study of
healthy twin pairs has shown that classical small-world
metrics on brain functional networks derived from eeG
data have high heritability 130, a necessary prerequisite for
their candidacy as disease endophenotypes. another
study of graph theoretical measures of anatomical networks derived from inter-regional correlations in cortical-thickness mRI measurements on a sample of normal
twins, singletons and singleton siblings of twins showed
that genetically determined frontoparietal networks had
small-world properties131. network metrics are arguably
more attractive as intermediate phenotypes than local
measures of brain (dis)organization, because computational models of network development are often available to test mechanistic hypotheses for how an observed
profile of anatomical or functional dysconnectivity in a
mature network might have been generated by earlier
developmental abnormalities24,132.
another example of how empirical and computational
approaches can be usefully combined is provided by studies that have ‘lesioned’ anatomical or functional network
models — for example, by deleting nodes or connections
— to explore how acute and focal damage could affect the
overall performance of brain networks70,133,134. networks
can be lesioned by random deletion of nodes or edges,
or by targeted attack on the highest-degree nodes in the
network. The vulnerability of the network to damage is
assessed by comparing its topological or dynamical behaviour after the lesioning to its intact behaviour. Different
network topologies confer different vulnerabilities to the
effects of random or targeted attack. For example, scalefree networks are robust to random error but highly
vulnerable to deletion of the network hubs. Brain functional networks with an exponentially truncated power
law degree distribution were found to be less vulnerable
to attack than scale-free networks70. In an anatomically
informed computational model, deletion of hub nodes
produced widespread disruptions of functional connectivity 53,134 that were consistent with effects reported in
focal human brain lesions135,136. computational lesioning
of network models was also used to explore the functional consequences of a gradual and precisely specified
disease process: the elimination of long-range projections
and the sprouting of short-range connections in a model
of epileptogenesis in the rat dentate gyrus137. The topology of the normal or non-epileptic dentate gyrus became
relatively over-connected and dynamically hyperexcitable
as a result of cellular changes previously described in relation to temporal lobe epilepsy. Other studies of models
of temporal lobe epilepsy have shown loss of small-world
topology in cellular networks during hypersynchronized
bursting 138 and have shown that variation of small-world
topological and synaptic properties of a computational
model can cause transitions between normal, bursting
and seizing behaviours139.
It is also conceivable that network analysis can be
used to further our understanding of the therapeutic
effects of pharmacological or psychological therapies.
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Dopaminergic drugs can modulate measures of functional
connectivity in animal and human fmRI140,141 and meG
recordings142, suggesting that drug effects might be quantifiable in terms of altered functional network topology.
This has been confirmed directly in a study which demonstrated that a dopamine D2 receptor antagonist impaired
the economical small-world properties of human brain
fmRI networks75. Future work might include efforts
using graph theoretical measures to quantify how therapeutically effective treatments remediate topologically
sub-optimal network configurations in patients.
Conclusions and prospects
It is clear that certain aspects of the organization of complex brain networks are highly conserved over different
scales and types of measurement, across different species
and for functional and anatomical networks. The archetypal brain network has a short path length (associated
with high global efficiency of information transfer), high
clustering (associated with robustness to random error),
a degree distribution compatible with the existence of
hubs, and a modular community structure. Furthermore,
anatomical networks are sparsely connected, especially
between nodes in different modules, and the ‘wiring
length’ (the physical distance that connections span) is
close to minimal. This profile of topological and geometric properties is typical not just of brain networks but
also of many other complex networks, including transport systems and intracellular signalling pathways17,73.
Why might this be so?
a parsimonious hypothesis is that many spatially
embedded complex networks have evolved to optimize
the same set of competitive selection criteria — high efficiency of information transfer between nodes at low connection cost — or to achieve an optimal balance between
functional segregation and integration that yields high
complexity dynamics14. If wiring cost was exclusively
prioritized the network would be close to a regular lattice, whereas if efficiency was the only selection criterion
the network would be random. The existence of a few
long-range anatomical connections can deliver benefits
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Acknowledgements
E.B. was supported by a Human Brain Project grant from the
National Institute of Mental Health and the National Institute
of Biomedical Imaging & Bioengineering. The Behavioural &
Clinical Neurosciences Institute in the University of Cambridge
is supported by the Wellcome Trust and the Medical Research
Council (UK). O.S. was supported by the JS McDonnell
Foundation.
Competing interests statement
The authors declare competing financial interests: see web
version for details.
FURTHER INFORMATION
Ed Bullmore’s homepage: http://www.neuroscience.cam.
ac.uk/directory/profile.php?edbullmore
Olaf Sporns’ homepage: http://www.indiana.edu/~cortex
Brain Connectivity Toolbox: http://www.brainconnectivity-toolbox.net/
BrainwaveR Toolbox: http://www.nitrc.org/projects/
brainwaver/
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